In HW 1 Problem 2, the square root of -1 is not to be viewed as i, j, or k which represents arbitrary (but fixed} elements satisfying their square in the quaterions is -1. Indeed in most quaternions, there are infinitely many elements satisying its square is -1. In particular, when viewing the quaternions over the complex numbers, do not assume i, j, or k is what we think of as the square root of -1 in the complex numbers. You only need to use equation after the (check!).